Zealots or not, they have to be consistent with their own narratives and motivations. It makes sense for Tri-Tach to use AI cores, hell, maybe even Hegemony, since it fits with them trying to be dominant over the sector, but not the Luddic Church or Path. The whole idea behind factions is that they have different ideologies, motives, and technologies to deal with whatever they face in the Sector.
Honestly? The church using cores they stole makes perfect sense because religious zealots DON'T MAKE SENSE. They make CONVERTS who DON'T QUESTION THEIR FAITH.
Now GET BACK TO CRUSADING.
EDITBy the way, had a thing happen to me where I was nonhostile to Hegemony and they were gonna do an AI inspection. Bribed them. Then later on they declare war on me. Next thing I know is they confiscated 13 Alpha cores. Declaring the war happened shortly before the inspection arrived, didn't seem to refund the 1 mil of bribes, yet it doesn't seem like my colony was resisting the inspection (generally, it's impenetrable to almost anything, since I have alpha cores plugged into heavy batteries and planetary shield, as well as max admin skills). Upon reloading the save prior to inspection arrival and them declaring war on me, it shows that the inspection forces are outmatched. Basically, if hegemony declares war on you just the moment before an inspection that you've bribed off, it seems like, instead of resisting, your colony complies with the inspection. Every other time they visited that colony, they weren't able to take anything.